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Ascendant0
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I'm currently taking Math Methods in Physics, and we're working on infinite series right now. In one of the examples, in order to find the limit, they do what you can see below in "Example 3"
As you can see, they turn the original equation into a ln equation, so that they can bring the exponent out in front, then change the sign so they can flip the ratio of the ln from "## 1/n ##" to "##n##". Then, they apply L'Hopital's rule and take the derivative to find the limit as that equation approaches infinity to be 0. Then, since they applied ln to the original function, they apply that value as an exponential value of e, which gives ## e^0=1##. That is an awful lot of changing an equation in multiple ways to get it where you want it to be, and I was surprised you can manipulate it to that extent, and still get the correct answer in the end.
So, what I'm wondering is if there is any limit as to how you can manipulate an equation like this to find the limit, so long as you apply the reverse of what you did to the end answer like they did? Can we do the same thing with any exponent e or logarithm to any limit? I mean I know adding certain things like multiplication or division and just reversing it at the end would deviate you from the correct answer (at least I believe it would), so I'm just trying to learn to what extent can we add things to the equation and just reverse it later like this, yet still get the correct limit in the end?
As you can see, they turn the original equation into a ln equation, so that they can bring the exponent out in front, then change the sign so they can flip the ratio of the ln from "## 1/n ##" to "##n##". Then, they apply L'Hopital's rule and take the derivative to find the limit as that equation approaches infinity to be 0. Then, since they applied ln to the original function, they apply that value as an exponential value of e, which gives ## e^0=1##. That is an awful lot of changing an equation in multiple ways to get it where you want it to be, and I was surprised you can manipulate it to that extent, and still get the correct answer in the end.
So, what I'm wondering is if there is any limit as to how you can manipulate an equation like this to find the limit, so long as you apply the reverse of what you did to the end answer like they did? Can we do the same thing with any exponent e or logarithm to any limit? I mean I know adding certain things like multiplication or division and just reversing it at the end would deviate you from the correct answer (at least I believe it would), so I'm just trying to learn to what extent can we add things to the equation and just reverse it later like this, yet still get the correct limit in the end?